Average Error: 30.2 → 0.4
Time: 1.6s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -4.3485033051676 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{2}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\\ \end{array}\]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -4.3485033051676 \cdot 10^{-310}:\\
\;\;\;\;-x \cdot \sqrt{2}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\sqrt{2}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\\

\end{array}
(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -4.3485033051676e-310)
   (- (* x (sqrt 2.0)))
   (* (sqrt (sqrt 2.0)) (* x (sqrt (sqrt 2.0))))))
double code(double x) {
	return sqrt((2.0 * x) * x);
}
↓
double code(double x) {
	double tmp;
	if (x <= -4.3485033051676e-310) {
		tmp = -(x * sqrt(2.0));
	} else {
		tmp = sqrt(sqrt(2.0)) * (x * sqrt(sqrt(2.0)));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -4.3485033051676e-310

    1. Initial program 29.9

      \[\sqrt{\left(2 \cdot x\right) \cdot x}\]

    2. Taylor expanded around -inf 0.4

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{2}\right)}\]

    3. Simplified0.4

      \[\leadsto \color{blue}{-x \cdot \sqrt{2}}\]

    if -4.3485033051676e-310 < x

    1. Initial program 30.4

      \[\sqrt{\left(2 \cdot x\right) \cdot x}\]

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{x \cdot \sqrt{2}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt_binary640.6

      \[\leadsto x \cdot \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)}\]

    5. Applied associate-*r*_binary640.4

      \[\leadsto \color{blue}{\left(x \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3485033051676 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{2}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020274 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))